The variational inequality is firstly derived. Compared to the work in [1]}, besides the vital first-order adjoint equation with Poisson measure, we can obtain four new adjoint equations, however, the second-order adjoint equation is a new BSDEP. Then we derive a new partially observed global maximum principle. More importantly, the maximum principle can reduce to the work [2]} when the forward-backward system with random jumps degenerate into the forward one without random jumps. Different from the framework in [2]} where the observed process \(Y\)  is viewed as a Brownian motion at the beginning, we not only consider a more natural framework([4]},[5]}), but also prove the equivalence between the two frameworks. More details can be seen in Remark REF , Remark REF , Remark REF .
